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Sommerfeld Theory of Metals ­ a quantum theory of independent free electrons Physical picture: Focus on one electron. Assume all other charges in the system are smeared out into a static neutral background (atomic details, fluctuations, & correlations ignored). All electrons are "equal." A simplified jellium model. Main difference from Drude model ­ Fermi Dirac statistics. Consider an electron gas in a cubic box of volume V = L3 . Two commonly used boundary conditions: 1. Infinite well, = 0 outside V; standing wave solutions; convenient for some problems. 2. Periodic boundary condition (PBC), or Born-von Karman BC:

x L, y , z x, y L, z x , y , z L x , y , z

1D example:

Wave function repeats.

We use PBC ­ convenient for mode counting. It is unphysical in 2D and 3D, but bulk properties are independent of V and boundary conditions as V .

1 Plane wave solution: k r V exp ik r

k = wave vector; quantum number

p k r i k r k k r

H k r

It is a momentum eigenfunction.

2 k 2 p2 k r k r k k r 2m 2m

k

2 k 2 2m

(dispersion relation)

PBC: exp ik x x L ik y y ik z exp ik x x ik y y ik z z

exp ik x L exp ik y L exp ik z L 1

k x, y ,z 2 nx , y , z L

etc.

nx , y , z integers

A dense grid of points in k space.

3 2 8 Volume each k point L V 3

V Density of points in k space = 8 3

k

V dk 8 3

for V

1 exp ik r k r Adding spin, V

k ,s

2V dk 8 3

Ground state at T = 0 (minimum energy) based on Pauli exclusion principle:

2 2 g A 0,0,0 0,0,0 ,0,0 ,0,0 L L 2 2 2 2 L ,0,0 L ,0,0 0, L ,0 0, L ,0 ...

A = Antisymm operator

g = a Slater determinant

Total energy is a minimum occupied k values are enclosed in a sphere in k space (Fermi sphere; Fermi surface; radius = Fermi wave vector)

N

k k F

2 8

3

2V

dk

k k F

2V 4 3 V 3 kF 2 kF 3 8 3 3

n

N k3 F2 V 3

k F 3 2 n

1

3

~ 1 Å-1 for typical metallic density.

F

TF

F

2 2kF (Fermi energy) ~ 1.5 ­ 15 eV, typically. 2m

kB

(Fermi temperature) ~104 ­ 105 K, typically (>> RT).

vF 2 F m (Fermi velocity) ~108 cm/s, typically (relativistic effects small).

Ground state properties at T = 0

2V Total energy E k 2 k 8 3 k

F

k kF

2V 3 2k 2 3 2k 2 d k 3 4 k 2 dk N F 2m 8 k kF 2m 5

E 3 (average energy per particle) F N 5

From first law of thermodynamics, TdS dU pdV dN 0 2 2 E 3 E 3 p NF N 3 2 n 3 ... 2 V 2 n F (not measurable) 3 5 V N V 5 V 5 2 m

p 2 Bulk modulus B V V 3 n F ~1010 ­ 1012 dyne/cm2, typically; about the right order of magnitude for metals.

Properties at T > 0 Fermi Dirac distribution (probability of occupancy)

0 f 1 (exclusion principle) f , T T exp 1 kT 1

T chemical potential = Gibbs free energy per particle; determined by normalization f k N

k,s

1 for 0 f , T 0 f0 0 for 0

f0

0 F

1

For T > 0, f ( , T )

1 2

F

kT F ~ 0.01 at RT for typical metals.

f(T) 1 1/2

~kT

OK to assume kT F (low temperature limit)

f ( , T ) ~ broadened negative step function, width ~kT

d f ( , T ) ~ broadened delta function, width ~kT d kT 2 T F O F ~ F for free electrons in 3D, to be verified (different for 2D & 1D). F

Density of states (levels) (per unit volume) g 1 g d # of one-electron levels between and d V 1 Very useful for calculating F k F g d , where F is a function of energy. V k ,s 1 Summation in k space converted into an integral over energy. d g V k ,s 1 2V 1 1 dk d 3 k 3 4 k 2 dk 2 k 2 d 3 V 8 4 d k ,s 2 k 2 d 2k For free electrons, . dk m 2m 1 V

1 V

k ,s

1

2

m k d d g 2 2

g

mk 2 m3 2 3 2 2

g F

mk F density of states at the Fermi level, relevant to many properties. 2 2

g f ,T

T=0

g f ,T

T>0

F

n

F N f , T g d g d = area under the curve, independent of T. 0 0 V This constraint yields T . Pictorially, it is clear that must be slightly less than F in order to conserve the area.

Define g 0 for 0 . Define

g d G .

n

d f , T g d G f , T G f , T d d

1 2 d Expansion: n G g g ... f , T d 2 d

1st term = G f d G f f G 2nd term = g 3rd term

f d 0 , because the integrand is odd about .

5th term O T 4 ; ...

2 2

6 6

kT

2

g ; 4th term = 0 (odd integrand);

2

n G

kT

g O T 4 G F G

(1)

n T 0 n T G G F

2

6

kT

2

g O T 4

2

6

kT 2 g O T 4

1 2

2

Expand:

F g F F

2

It is clear that F O T

6 if only the first term is kept. The second term O T 4 .

g F

2

kT

2

g O T 4

F

2

6

kT

2

g F g F

O T 4 F O T 2

Whew! (Sommerfeld expansion)

Specific heat (at constant volume)

TdS dQ dU pdV

c

1 dQ 1 U u V dT V V T V T V

Repeat the above derivation with g g , see Eq. (1) previous page.

u c

g f , T d g d

0

2

6

kT

2

g g O T 4

u d 2 2 2 d 2 3 g k T g g 6 kT 2 g g dT O T T dT 3 2 2 g F 2 2 O T 3 c g k T k T g g O T 3 g F 3 3

2 g F 2 2 c F g F k 2T k T g F F g F O T 3 g F 3 3

c

2

3

k 2T g F O T 3 mk F , 2 2

With g F

c

2 kT nk 2 F

Cf. cDrude

3 nk 2

cS kT 0.01 at R.T. cD F

For metals at RT, c is dominated by phonon contributions: c phonon 3 nk (verified later). Below a few K, c is dominated by electronic contributions (phonons are quenched out). Results in fairly good agreement with experiment (within 2x, typically) Why is cS cD ?

f(T) 1 1/2

kT Number of electrons excited n F Energy for each excited electron kT

~kT

kT un

F

2

kT So, c F

nk

For the Drude model, each electron has thermal energy kT . So,

cS kT 0.01 cD F

Consequences of different statistics: Physical Properties Volume specific heat c DC conductivity o AC conductivity Dielectric function Hall coefficient RH Magnetoresistance B Thermal power Q v2 Drude 3 nk 2

ne2 m o 1 i 1 4 i 1 nec none

Sommerfeld 2 kT nk 2 F Same* Same* Same* Same* Same* Same* 2 2 vF F m ~108 cm/s ~100 Å Same*

Ratio (D/S) ~100 at RT 1 1 1 1 ~100 at RT ~0.01 at RT ~0.1 at RT ~0.1 at RT ~1 (cancellation)

c / 3ne 3kT / m

~107 cm/s v v (with deduced from ~10 Å Thermal conductivity K v2 c 3

*Formulas are the same as those for the Drude results because we can still use the same d p ave p basic equation ave f dt = phenomenological relaxation time = relaxation time for distribution function f to relax back to f o

Fermi sphere

Fermi sphere displaced by an electric field

Displacement in the steady state with a field: p0 f ave Removing the field,

= time to return to initial distribution

p ave p0 exp t / ave

This picture illustrates that n in

kT ne 2 is the total electron density, not n . m F

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